Some classes of concatenated quantum codes: constructions and lower bounds

2015 ◽  
pp. 385-405
Author(s):  
Hachiro Fujita
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Polynomial Time ◽  
Coding Theory ◽  
Minimum Distance ◽  
Concatenated Codes ◽  
Quantum Codes ◽  
Decoding Algorithm ◽  
Quantum Analogue

In classical coding theory code concatenation is successfully used to construct good errorcorrecting codes and most of the asymptotically good codes known so far are based on concatenation. In this paper we present some classes of asymptotically good concatenated quantum codes, which are a quantum analogue of classical concatenated codes, and derive lower bounds on the minimum distance and the rate of the codes. Our bounds improve on the best lower bound of Ashikhmin–Litsyn–Tsfasman and Matsumoto for rates smaller than about one half. We also give a polynomial-time decoding algorithm for the codes that can decode up to one fourth of the lower bound on the minimum distance of the codes.

scholarly journals Augmenting the Power of (Partial) MaxSat Resolution with Extension

2020 ◽  
Vol 34 (02) ◽  
pp. 1561-1568 ◽  
Author(s):  
Javier Larrosa ◽  
Emma Rollon
Keyword(s):  
Lower Bound ◽  
Objective Function ◽  
Lower Bounds ◽  
Polynomial Time ◽  
Proof System ◽  
Extension Rule ◽  
Resolution Proof

The refutation power of SAT and MaxSAT resolution is challenged by problems like the soft and hard Pigeon Hole Problem PHP for which short refutations do not exist. In this paper we augment the MaxSAT resolution proof system with an extension rule. The new proof system MaxResE is sound and complete, and more powerful than plain MaxSAT resolution, since it can refute the soft and hard PHP in polynomial time. We show that MaxResE refutations actually subtract lower bounds from the objective function encoded by the formulas. The resulting formula is the residual after the lower bound extraction. We experimentally show that the residual of the soft PHP (once its necessary cost of 1 has been efficiently subtracted with MaxResE) is a concise, easy to solve, satisfiable problem.

scholarly journals From Generalization of Bacon-Shor Codes to High Performance Quantum LDPC Codes

2021 ◽  
Author(s):  
Jihao Fan ◽  
Jun Li ◽  
Ya Wang ◽  
Yonghui Li ◽  
Min-Hsiu Hsieh ◽  
...  
Keyword(s):  
Minimum Distance ◽  
High Performance ◽  
Ldpc Codes ◽  
Quantum Error Correction ◽  
Concatenated Codes ◽  
Quantum Codes ◽  
Error Correction Codes ◽  
Distance Bound ◽  
Quantum Error ◽  
Minimum Distance Bound

Abstract We utilize a concatenation scheme to construct new families of quantum error correction codes that include the Bacon-Shor codes. We show that our scheme can lead to asymptotically good quantum codes while Bacon-Shor codes cannot. Further, the concatenation scheme allows us to derive quantum LDPC codes of distance Ω(N2/3/loglogN) which can improve Hastings’s recent result [arXiv:2102.10030] by a polylogarithmic factor. Moreover, assisted by the Evra-Kaufman- Zémor distance balancing construction, our concatenation scheme can yield quantum LDPC codes with non-vanishing code rates and better minimum distance upper bound than the hypergraph product quantum LDPC codes. Finally, we derive a family of fast encodable and decodable quan- tum concatenated codes with parameters Q = [[N,Ω(√N),Ω(√N)]] and they also belong to the Bacon-Shor codes. We show that Q can be encoded very efficiently by circuits of size O(N) and depth O(√N), and can correct any adversarial error of weight up to half the minimum distance bound in O(√N) time. To the best of our knowledge, they are the most powerful quantum codes for correcting so many adversarial errors in sublinear time by far.

TIME-OPTIMAL DIGITAL GEOMETRY ALGORITHMS ON MESHES WITH MULTIPLE BROADCASTING

1995 ◽  
Vol 09 (04) ◽  
pp. 601-613
Author(s):  
V. BOKKA ◽  
H. GURLA ◽  
S. OLARIU ◽  
J.L. SCHWING ◽  
I. STOJMENOVIĆ
Keyword(s):  
Lower Bound ◽  
Convex Hull ◽  
Lower Bounds ◽  
Minimum Distance ◽  
Fundamental Problem ◽  
Optimal Solution ◽  
The Other ◽  
Digital Geometry ◽  
Black Pixel ◽  
Time Optimal

The main contribution of this work is to show that a number of digital geometry problems can be solved elegantly on meshes with multiple broadcasting by using a time-optimal solution to the leftmost one problem as a basic subroutine. Consider a binary image pretiled onto a mesh with multiple broadcasting of size [Formula: see text] one pixel per processor. Our first contribution is to prove an Ω(n1/6) time lower bound for the problem of deciding whether the image contains at least one black pixel. We then obtain time lower bounds for many other digital geometry problems by reducing this fundamental problem to all the other problems of interest. Specifically, the problems that we address are: detecting whether an image contains at least one black pixel, computing the convex hull of the image, computing the diameter of an image, deciding whether a set of digital points is a digital line, computing the minimum distance between two images, deciding whether two images are linearly separable, computing the perimeter, area and width of a given image. Our second contribution is to show that the time lower bounds obtained are tight by exhibiting simple O(n1/6) time algorithms for these problems. As previously mentioned, an interesting feature of these algorithms is that they use, directly or indirectly, an algorithm for the leftmost one problem recently developed by one of the authors.

scholarly journals On Third-Order Nonlinearity of Biquadratic Monomial Boolean Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Brajesh Kumar Singh
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Boolean Function ◽  
Lower Bounds ◽  
Boolean Functions ◽  
Coding Theory ◽  
Block Ciphers ◽  
Covering Radius ◽  
Third Order ◽  
Complicated Problem

The rth-order nonlinearity of Boolean function plays a central role against several known attacks on stream and block ciphers. Because of the fact that its maximum equals the covering radius of the rth-order Reed-Muller code, it also plays an important role in coding theory. The computation of exact value or high lower bound on the rth-order nonlinearity of a Boolean function is very complicated problem, especially when r>1. This paper is concerned with the computation of the lower bounds for third-order nonlinearities of two classes of Boolean functions of the form Tr1nλxd for all x∈𝔽2n, λ∈𝔽2n*, where a d=2i+2j+2k+1, where i, j, and   k are integers such that i>j>k≥1 and n>2i, and b d=23ℓ+22ℓ+2ℓ+1, where ℓ is a positive integer such that gcdℓ,𝓃=1 and n>6.

scholarly journals Minimum distance functions of complete intersections

2018 ◽  
Vol 17 (11) ◽  
pp. 1850204 ◽  
Author(s):  
José Martínez-Bernal ◽  
Yuriko Pitones ◽  
Rafael H. Villarreal
Keyword(s):  
Lower Bound ◽  
Complete Intersection ◽  
Polynomial Ring ◽  
Distance Function ◽  
Coding Theory ◽  
Minimum Distance ◽  
Distance Functions ◽  
Complete Intersections ◽  
Dimension One ◽  
Sharp Lower Bound

We study the minimum distance function of a complete intersection graded ideal in a polynomial ring with coefficients in a field. For graded ideals of dimension one, whose initial ideal is a complete intersection, we use the footprint function to give a sharp lower bound for the minimum distance function. Then we show some applications to coding theory.

A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers

10.37236/1188 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Ramsey Numbers

For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.

scholarly journals A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 164
Author(s):  
Tobias Rupp ◽  
Stefan Funke
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real World ◽  
Road Network ◽  
Upper And Lower Bounds ◽  
Bounded Degree ◽  
Shortest Path Routing ◽  
Graph Classes ◽  
Speed Up ◽  
To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

scholarly journals On Computing Multilinear Polynomials Using Multi- r -ic Depth Four Circuits

2021 ◽  
Vol 13 (3) ◽  
pp. 1-21
Author(s):  
Suryajith Chillara
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Complexity Measure ◽  
Natural Question ◽  
Multilinear Polynomial ◽  
Arithmetic Circuit ◽  
Theory Of Computing ◽  
Small Constant ◽  
Multilinear Polynomials ◽  
The Individual

In this article, we are interested in understanding the complexity of computing multilinear polynomials using depth four circuits in which the polynomial computed at every node has a bound on the individual degree of r ≥ 1 with respect to all its variables (referred to as multi- r -ic circuits). The goal of this study is to make progress towards proving superpolynomial lower bounds for general depth four circuits computing multilinear polynomials, by proving better bounds as the value of r increases. Recently, Kayal, Saha and Tavenas (Theory of Computing, 2018) showed that any depth four arithmetic circuit of bounded individual degree r computing an explicit multilinear polynomial on n O (1) variables and degree d must have size at least ( n / r 1.1 ) Ω(√ d / r ) . This bound, however, deteriorates as the value of r increases. It is a natural question to ask if we can prove a bound that does not deteriorate as the value of r increases, or a bound that holds for a larger regime of r . In this article, we prove a lower bound that does not deteriorate with increasing values of r , albeit for a specific instance of d = d ( n ) but for a wider range of r . Formally, for all large enough integers n and a small constant η, we show that there exists an explicit polynomial on n O (1) variables and degree Θ (log 2 n ) such that any depth four circuit of bounded individual degree r ≤ n η must have size at least exp(Ω(log 2 n )). This improvement is obtained by suitably adapting the complexity measure of Kayal et al. (Theory of Computing, 2018). This adaptation of the measure is inspired by the complexity measure used by Kayal et al. (SIAM J. Computing, 2017).

scholarly journals Lower bounds for the state complexity of probabilistic languages and the language of prime numbers

2020 ◽  
Vol 30 (1) ◽  
pp. 175-192
Author(s):  
NathanaËl Fijalkow
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Prime Numbers ◽  
Regular Languages ◽  
Automata Theory ◽  
Seminal Paper ◽  
Probabilistic Automata ◽  
State Complexity ◽  
Probabilistic Languages ◽  
Alternating Automata

Abstract This paper studies the complexity of languages of finite words using automata theory. To go beyond the class of regular languages, we consider infinite automata and the notion of state complexity defined by Karp. Motivated by the seminal paper of Rabin from 1963 introducing probabilistic automata, we study the (deterministic) state complexity of probabilistic languages and prove that probabilistic languages can have arbitrarily high deterministic state complexity. We then look at alternating automata as introduced by Chandra, Kozen and Stockmeyer: such machines run independent computations on the word and gather their answers through boolean combinations. We devise a lower bound technique relying on boundedly generated lattices of languages, and give two applications of this technique. The first is a hierarchy theorem, stating that there are languages of arbitrarily high polynomial alternating state complexity, and the second is a linear lower bound on the alternating state complexity of the prime numbers written in binary. This second result strengthens a result of Hartmanis and Shank from 1968, which implies an exponentially worse lower bound for the same model.

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